A278963 - OEIS

%I #11 Apr 29 2019 08:24:33

%S 1,1,2,3,4,2,6,6,8,6,10,7,12,6,8,14,16,14,18,12,14,14,22,12,24,18,24,

%T 18,28,11,30,28,26,30,26,28,36,30,30,27,40,20,42,30,32,30,46,32,48,42,

%U 32,38,52,36,46,43,50,42,58,32,60,30,52,60,50,48,66,60,50

%N a(n) is the number of k, 1<=k<=n, such that gcd(n,k) divides binomial(n,k).

%C Length of row n of A278961.

%C a(n) is odd if and only if n = 1 or n is in A048618.

%C a(n) <= n-1 for n>1.

%C a(n) = n-1 if n is a prime or the square of a prime.

%C a(n) >= A000010(n).

%H Robert Israel, <a href="/A278963/b278963.txt">Table of n, a(n) for n = 1..10000</a>

%e a(8) = 6 because gcd(8,k) divides binomial(8,k) for k=1,2,3,5,6,7 but not k=4 or k=8.

%p f:= proc(n,m) if binomial(n,m) mod igcd(n,m) = 0 then m else NULL fi end proc:

%p [seq(nops([seq(f(n,m),m=1..n)]),n=1..200)];

%t a[n_] := Sum[Boole[Divisible[Binomial[n, k], GCD[n, k]]], {k, 1, n}];

%t Array[a, 100] (* _Jean-François Alcover_, Apr 29 2019 *)

%o (PARI) a(n) = sum(k=1, n, (binomial(n, k) % gcd(n, k))==0); \\ _Michel Marcus_, Dec 04 2016

%Y Cf. A000010, A048618, A278961.

%K nonn

%O 1,3

%A _Robert Israel_, Dec 02 2016